L(s) = 1 | + (−0.889 + 2.68i)2-s + (−6.41 − 4.77i)4-s − 14.9i·5-s + 30.0i·7-s + (18.5 − 12.9i)8-s + (40.0 + 13.2i)10-s + 55.9·11-s + 57.4·13-s + (−80.6 − 26.7i)14-s + (18.3 + 61.2i)16-s + 29.2i·17-s − 0.709i·19-s + (−71.2 + 95.7i)20-s + (−49.7 + 150. i)22-s + 48.0·23-s + ⋯ |
L(s) = 1 | + (−0.314 + 0.949i)2-s + (−0.802 − 0.596i)4-s − 1.33i·5-s + 1.62i·7-s + (0.818 − 0.573i)8-s + (1.26 + 0.419i)10-s + 1.53·11-s + 1.22·13-s + (−1.54 − 0.510i)14-s + (0.287 + 0.957i)16-s + 0.417i·17-s − 0.00856i·19-s + (−0.796 + 1.07i)20-s + (−0.482 + 1.45i)22-s + 0.435·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.23071 + 0.618331i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23071 + 0.618331i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.889 - 2.68i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 14.9iT - 125T^{2} \) |
| 7 | \( 1 - 30.0iT - 343T^{2} \) |
| 11 | \( 1 - 55.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 57.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 29.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 0.709iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 48.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 172. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 45.2iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 248.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 51.3iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 19.9iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 10.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 37.0iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 411.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 308.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 113. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 728.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 487. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 1.16e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.19e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 624.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.40251362112317424671557863894, −12.48491097493491093304948630127, −11.45612336851587989447590878000, −9.523804529324554107851887053826, −8.861334583392060324720967740624, −8.283984351198882842652115313165, −6.37114175875980394492901795507, −5.57522890536689878188746700122, −4.23111620663900938470190089629, −1.29542689310078805923529202204,
1.17635159623305234534222665133, 3.29528533310982126948121642415, 4.14525769418205933127639072043, 6.57166332097891505668337804537, 7.48041932658900336469057391496, 9.005931385934354674551698600766, 10.19805861149792697396634786431, 10.94815778724127833402753237546, 11.57328212094655964903194171688, 13.14967250975389923342370726059